Apparatus and method for producing square-law function



March l0, 1970 APPARATUS AND METHODFO PRODUCING SQUARE-LAW FUNCTIONFiled Aug. 27. 1965 4 Sheets-Sheet 1 Jerry M. Collings Attorneys March10, 1970 J. M.- coLLlNGs APPARATUS AND METHOD FOR `PRODUCING SQUARE-LAWFUNCTION- Filed Aug. 2v. i965 4 Sheets-Sheet 2 2G Fig. 2d

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BY Jerry M. Collings atm, WL 0( u Attorneys March 10, 1970 .1.M.coLLlNGs 3,500,445

I APPARATUS AND METHOD FOR PRODUCING 4SQUARE-LAW FUNCTION Filed Aug. 27.196s 4 sheets-sheet s ERROR CURVE ,107" Fig. 4a

Jerry M. Collings BY WI M ab *01km .La Attorneys J'. M, CoLLlNGs3,500,445

APPARATUS AND METHOD FOR PRODUCING SQUARE-LAW FUNCTION Filed Aug. 27.1965 4 sheets-sheet 4 March 10, 1970 .....udvhw www.

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United States Patent O 3,500,445 APPARATUS AND METHOD FOR PRODUCINGSQUARE-LAW FUNCTION Jerry M. Collings, Concord, Calif., assignor toZeltex, Inc., a corporation of California Filed Aug. 27, 1965, Ser. No.483,180 Int. Cl. G06f 15/34 U.S. Cl. 23S-197 13 Claims ABSTRACT OF THEDISCLOSURE In an analogmultiplier, method and apparatus are disclosedfor generating a voltage output signal having a segmental formapproximating a parabolic of squared function of an input voltagesignal. The input signal is shaped by a series of non-linear electricalconversion networks into a plurality of half wave triangular Voltagesignals having pre-selected frequency, amplitude and symmetry with theinput voltage and thereafter all of the triangular voltage signals areelectrically summated to produce the desired segmental parabolic orsquared output signal.

The invention relates to electronic analog multipliers of aquarter-square type and more particularly to that portion of theapparatus which performs the squaring operation. An analog device ashere used represents the output as a voltage.

The quarter-square type analog multiplier has as its basis of operationthe algebraic relationship:

The basic scheme of the quarter-square multiplier is shown in FIGURE lof the drawings. The principal parts of this type of multiplier are (l)one each summing and difference input networks for two-phase inputsignals which form the quantities a(xly) and y-a(x-y); (2) one eachpositive and negative absolute value networks which select only positivevalues of a(x-iy) and negative values of -a(xy); (3) one each positiveand negative square law function generators to form the quantitiesb(xl-y)2 and -b(x-y)2; and (4) an output summing unit which forms thedesired output b(x|y)2-b(x-y)2=cxy.

Existing quarter-square multipliers generally employ passive diodefunction generators to perform the squaring operation. The majordisadvantages of this-method are temperature sensitivity, cumulativeerror caused by leakage current in the many diodes needed to'achievehigh accuracy, and the ditliculty in adjusting the networks within thediode function generators to obtain the desired overall output curvecharacteristics because of the uncertainty of the individual diodeswitching characteristics.

An objectof the present invention is to provide an active-type squaringnetwork which to a very large degree eliminates the various errorsassociated with the diodes in the passive type squaring network and thusto obtain greater precision, accuracy and dependability.

Another object of the present invention is to provide an apparatus andmethod for producing the desired square-law function having the improvedresults above and to do so at a reasonable cost.

A further object of the present invention is to provide an apparatus andmethod for producing a square-law function of the character above andwhich will square any signal either DC or AC and do it substantiallyinstantaneously whereby the apparatus .will be able to constantlypresent the square of an input voltage which may vary rapidly with time;and which will automatically convert a negative input voltage to apositive squared output voltage in accordance with the algebraicprinciple that the square of a negative quantity is always a positivequantity.

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The invention possesses other objects and features of advantage, some ofwhich of the foregoing will be set forth in the following description ofthe preferred form of the invention which is illustrated in the drawingsaccompanying and forming part of this specification. It is to beunderstood, however, that variations in the showing made by the saiddrawings and description may be adopted within the scope of theinvention as set forth in the claims.

Referring to said drawings:

FIGURE 1 is a schematic block diagram of a quartersquare analogmultiplier.

FIGURE 2a is a graph showing one of the half-wave rectied triangularwave functions used in the present method and apparatus.

FIGURE 2b is a graph showing another of the triangular wave functions.

FIGURE 2c is a graph of another of the triangular wave functions.

FIGURE 2d is a graph of another of the triangular wave functions.

FIGURE 2e is a graph showing a segmental approximation of a parabolicfunction derived by summating the functions shown in FIGURES 2a, 2b, 2cand 2d.

FIGURE 3 is a graph showing an error curve function used in the presentapparatus and method.

FIGURE 4 is a schematic block diagram showing essential portions of theapparatus and method used for generating and summating the particularwave form functions used in the present invention.

FIGURE 5 is a schematic diagram of a quarter-square electronic analogmultiplier constructed in accordance with the present invention.

The analog multiplier is a device in which a voltage x and a voltage yare applied as inputs and from which a voltage cxy appears as output.Commonly these are volt multipliers having a scale factor c=-0.01. It isusual for each x and y to vary as functions of time; and an importantcharacteristic of a present multiplier is its ability to multiplyfaithfully when the x and the y inputs change rapidly in value.

With reference to FIGURE l the two input voltages x and y are applied tosumming networks 11 and 12 and difference networks 13 and 14. Voltagesrepresenting minus x and minus y must be provided from a source externalto the amplifier and these are likewise applied to the networks of 12,13 and 14 as illustrated. The outputs of these networks as noted onFIGURE l are applied to absolute value networks 16 and 17 which provideoutputs of a|x+y| and -a|x-`y|. These two outputs are then fed intosquaring circuits 18 and 19, the outputs from which are represented bythe products b(x-ly)2 and-b(xy)2. The latter outputs are then fed into asumming network 20 the output of which is the desired product cxy inaccordance with well understood algebraic addition.

The present invention is directed to the squaring networks 18 and 19. Inthe quarter-square analog multipliel the input voltages to thesesquaring networks are alx+y| and -a|x-y|. For simplicity in thedescription that follows only one squaring circuit will be described andthe input signal will be identied as S; it being noted that reverseddiodes in the lower squaring circuit 19 retain the negative sign of thesignal being squared in that circuit. The generation of kS2 will now bediscussed.

In a squaring network a'n input voltage is converted to an outputvoltage which is the square of an input voltage multiplied by anappropriate constant. Ideally the relationship starts at zero so that ifthe input voltage is zero the output voltage is likewise zero so that adirect reading output is obtained without adjustment. If this curve wereplotted, say on the face of an oscilloscope, with Em as the abscissa andEout as the ordinate, the

curve would appear. as a parabola. In a larger sense, therefore, thefunction of the present invention is to provide electronically aparabolic function; and this is accomplished in accordance with thepresent invention by summating a plurality of symmetrical half waverectified triangular wave functions, shown in FIGURE 2 as S1, S2, S3 andS1, having a series relationship in which the successive functions havea frequency progression of 2j and with the base break point of eachtriangular function coinciding with the peak of the next higher orderseries function. If we let p equal the number of these triangular wavesas illustrated in FIGURES 2a, 2b, 2c and 2d, the summating of thesetriangular waves will produce a segmental curve as seen in FIGURE 2ehaving 2P segments or break points. Finally in accordance with thepresent invention the successive peak magnitudes of the severaltriangular wave functions follow a relationship producing a change inslope in successive segments wherein the change in slope is a constant.The result is a segmental approximation of a parabolic function.

The foregoing may -be seen graphically by an examination of FIGURES 2a,2b, 2c, 2d and 2e. With reference to FIGURE 2a, it will be seen that thefunction S4 is plotted as the ordinate and 11 as the abscissa and thelatter has been subdivided for convenience into sixteen units. S1appears at 11:1 as the beginning of an inclined ramp which extends to11:2 and then breaks downwardly to zero at 11:3. In accordance with therectified portion of the definition of this function, no signal appearsbetween 11=3 to n:5. At 11:5 the function repeats as above and similarlyrepeats at 11:9 and 11:13. In other words if this were a full wavefunction there would be a negative portion of the curve appearingbetween 11:3-5, 11:7-9, and 11:11-13. Because of the rectification, thisnegative portion of the function is removed. The peak amplitude of thefunction S1 is set for purpose of this illustration at 1.

With reference to FIGURE 2b it will be seen that a similar symmetricalhalf wave rectified triangular wave function S3 starts at zero value at11:2 and rises as a straight ramp to 11:4 then extends downwardly tozero at n=6; and again repeats at 11:10, with no signal appearingbetween 11:0-2, 11:6-10, and 11:14-16. S3 represents the next lowerorder function of the series relationship and accordingly there is afrequency progression between S3 and S4 of 2, that is in the presentillustration S3 has two half cycles and S4 has four half cycles. It willalso be seen that the base points viz 11:2, 6, 10 andl 14 of function S3coincide with the peaks of the next higher order series function S1. Thepeak magnitudes of function S3 is set at 6.

With reference to FIGURE 2c it will be seen that the function S2 is zerofrom n=0 to 11:4, then rises on a ramp to 11:8 and decreases in asimilar ramp to zero at n1:12 and remains at zero for the balance of theabscissa n=12 to 111:16. This function S2 follows the relationship abovedefined in that its frequency is one-half of the next higher orderfunction S3; and its base break points 4 and 12 coincide with the peaksof S3. The peak of function vS2 is set at 28.

With reference to FIGURE 2d it will be noted that function S1 is zerofrom n:0 to 11:8 then rises as a straight ramp to n:16. If the curvewere continued the ramp would then start down so that function S1 incommon with functions S2, S3 and S4 is a symmetrical half wave rectifiedtriangular wave function, only a portion of which is here used. It willalso be observed that the frequency of S1 is one-half of S2 and the basebreak point of S1, viz., 11:8 is located at the peak of the next higherorder series function S2. The peak amplitude of |51 is adjusted to 120.

As will be apparent from the foregoing, the slope of the successivecurves S1-S4 decreases in the following order: 15, 7, 3 and 1. Thusviewing the slope progression in a reversed fashion, that is, readingfrom the higher to the lower order functions, the slope of eachsuccessive wave form increases by powers of two. The summation offunctions S1 plus S2 plus S3 plus S1 appears in FIG- URE 2e. Therelationship of these figures permits a graphic addition. From 11:0 to11:1 all functions are Zero and hence the summation shown in FIGURE 2eas segment 21 is likewise Zero. From 11:1 to 11:2 the only functionappearing is S4 which has a slope of 1. Hence the next segment 22 of thesummated function S is 1. Between 11:2 to 11:3 S4 subtracts its slope of1 from S3 whose slope is 3 so that the next segment 23 has a slope of 2.Between 11:3 to 11:4 only function S3 appears and accordingly thecorresponding segment 24 will have a slope of 3. Between 11:4 to 11:5,function S3 subtracts from function S2 so that the resulting segment 25has a slope of 7-3:4. From 11:5 to 11:6, function S4 is additive so thatthe resulting segment 26- has a slope of 7-3+1:5. From 11:6 to 11:7function S1 subtracts from function S2 so that the resulting segment 27has a slope of 7-1:6. From 111:7 to 11:8 only function S2 is involved sothat the corresponding segment 28 will have a slope of 7. Between 11:8and 11:9 function S2 subtracts from function S1 so that the resultingsegment 29 has a slope of 15-7:8. Between 11:9 and 111:10 function S4 isadded in so that the resulting segment 30 has a slope of 15-7-}1:9.Between 11:10 and 11:11 function S1 subtracts while function S3 adds sothat the slope of the resulting segment 31 is 15-7-|3-1:10. Between11:11 and 11:12 function S1 is at zero so that the resulting segment 32has a slope of 15-7-}-3:11. Between 11:12 and 11:13 function S3subtracts from function S1 so that the resulting segment 33 has a slopeof 15-3:12. Between 11:13 and 11:14 function S4 is added so that theresulting segment 34 has a slope of 15-3-|-1:13. Between 111:14 and11:15 Ifunction S4 is subtracted from function S1 so that the resultingsegment 35 has a slope of 15-1:14. Between 11:15 and 11:16 only functionS1 is involved so that the final segment 36 has a slope of 15.

The successive peaks or break points of the segments 21-36 will fall asseen in FIGURE 2e at 0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 79, 91,105, and 120.

For convenience in understanding the summation of 4a series ofsymmetrical half wave rectified triangular wave functions as aboveexplained the analysis of each segment 21-36 is presented in thefollowing table:

Summated Individual Slopes m- Peak 11 Segment S1=15 Sg=7 S3=3 S4=1 SlopeValue Ko, MKO

2 2D Where the first three terms represent the amplitudes of S1020),52(1'21), S3(j=2) respectively and the last term is a general formuladefining the amplitude of the S(j-|1)th triangular function for valuesof and where: l

The present invention accordingly may be used whenever it is desired togenerate a segmental approximation of a parabolic function having thegeneralized formula of AxZ-i-Bx-i-C. Where the invention is used for aquarter square multiplier a further refinement is desired viz. theelimination of the linear term Bx in the generalized formula or the teunn in the formula of the curve shown in FIGURE 2e. This may be done byshifting the base break points of the several triangular wave functionsS1, S2, S3 and S1 one-half of a unit of n to the left as of theirpositions seen in FIGURES 2a-2e.

In FIGURES 2a-2e, the horizontal coordinate or abscissa is representedby the variable n which in the present apparatus is the scaled inputvoltage. Once the number of triangular wave for-ms is selected and thenumber of segments or break points thus established, the

units on n may -be considered as the full-scale input voltage for whichthe system is designed, divided by the number of segments. In theillustration of FIGURES 2a-2e, four triangular wave forms are used thusproducing a segmental parabolic curve having 24 or 16 segments. It isconvenient to consider nmx to be the amount of input voltage which wouldcorrespond with the upper limit of the parabolic curve and which wouldbe the maximum input voltage for which the system is designed.Conveniently, the input voltage range may be set at 100 volts andaccordingly each unit of n in the illustration given would be 100divided by 16.

It can be shown that by the step of shifting all of the triangular wavesto the left by an amount of one-half unit, by adding a constant bias ofminus one-half unit to the input signal n, the function becomes S=1/2(n2-1A as illustrated in dashed lines in FIGURE 2e.

Summarizing the foregoing and restating portions of the definition insome alternate terms for clarity it can be stated that the straight-lineapproximation of the parabolic function illustrated in FIGURE 2e isgenerated as the sum of a series of half-wave symmetrical triangularwave functions S1, S2, S3, S4, etc., having the following relationships:

(l) The number of cycles of each waveform is 2P-2 where p is the orderof the waveform in the series.

(2) The horizontal shift from zero of each successive waveform, relativeto the maximum span of the variable n, is 11mm/2P (3) The slope of eachsuccessive waveform from the highest order down to S1 changes accordingto increasing powers of 2, forming a sequence proportional to 1:3:7:15:3l: etc. The relationship of the slopes is determined by therequirements that:

(a) the sum of the slopes at any Value of n must be equal to the slopeof the curve of the desired output function at the same value of n, and

(b) the slope of the function 1/2(n2-n) increases by a constant amountfor each successive unit increment of n. In the straight-lineapproximation of the function, the significant unit increment of n isthe constant value corresponding to the distance between breakpointsalong the output curve which occur at the junctions of the straight-linesegments.

Most importantly, in the foregoing it will be observed that the numberof segments in the parabolic function, and hence the accuracy of thesystem, is greatly multiplied over the number of basic circuits ortriangular wave forms used. Generally for p rectifying circuitsproviding p number of symmetrical half-wave rectified triangular wavefunctions, 2p segments in the parabolic function is obtained.

As hereinabove noted, the accuracy of the square-law curve simulated bythe foregoing method increases with the number of straight-line segmentscomprising the curve which in turn is dependent upon the number ofcomponent wave forms. Since the number of segments is equal to 2p wherep is the number of component wave forms, the method and apparatus of thepresent invention enables the production of a curve approximating aparabola to any degree of accuracy desired. From a practical standpoint64 segments approach an optimum result since such a curve has a finegranular error which is within the tolerances of the electricalcomponents used to generate the wave forms. For example the theoreticalerror with 64 sections is approximately 0.003 volt out of volts fullscale, which is 0.003 percent. In most analog devices, 0.01 percent isusually considered very good.

The generation of the triangular Wave function is not as simple as doinga similar type job in a straight passive manner because as will be morefully hereinafter shown, it requires a number of feed-forward amplifiersand sum- 'ming resistors for the several triangular wave functions.

Assuming it is desired to generate the 64-section parabola, it wouldtake six of these associated circuits. The last two or three of thesecircuits have quite a number of input resistors because all of theprevious units sum into the later sections. Accordingly, it is a featureof the present invention that the last several triangular wave functionsare replaced by a single special error correction wave function whichmay be obtained by a passive diode network. The present apparatus andmethod thus combines a basic output curve constructed of a relativelylow number of triangular wave forms with an error correction curve toproduce a final output curve consisting of a relatively large number ofsegments.

Examination of wave functions S1, S2, S3 and S4 in FIG- URES 2a-2d showsthat each successive wave form provides a correction to the functionapproximation generated by the sum of preceding wave forms. The truecorrection curve at any stage of the progression is equal to thedifference of the continuous square-law function and the sum of theprogression of triangular wave forms. This difference is itself arepetitive parabolic function.

As a feature of the present invetnion I have found that a satisfactoryerror curve may be generated by the fourth triangular Wave function S4.To do so, wave form 54 is generated as a full wave as shown in solidline 42, in FIG- URE 3 half Wave rather than half wave, as is fed into apassive diode network to obtain the parabolic error curve 41.

Method of generating component triangular wave forms A series ofhalf-Wave triangular wave forms which meets the requirements of thediscussion relating to FIGURE 2 can be generated by the methodillustrated in FIGURE 4. The first wave form of the series, S01 isgenerated by a signal summing-inverting-rectifying device N1. Thisdevice produces an output signal which is proportional to the inputsignal Sin, representing the variable n. N1 may be defined as anon-linear function generating electric circuit having input and outputrelationships as follows:

if input S 0, output=CS if input SSO, output=0 where C is a constant. Ahorizontal offset from the original signal shown in the box N1, isdepicted in FIGURE 4a and is determined by the value of input biassignal SB1. This bias signal (opposite in polarity to Sm) holds theoutput of the device N1 at zero -because of the rectifyingcharacteristic until the absolute value of input signal exceeds it. Theslope of the inverted output wave form S01 is controlled by theproportionality or gain characteristic of the device N1.

A second summing-rectifying-inverting device N2 is identical to N1except that (l) the bias signal SBZ is half the value of SBI, (2) theslope of the output wave form S02 is set at half the value of the slopeof S01, and (3) the inverted output S01 is added to the summing input 43of device N2. The resultant wave form is shown in FIG- URE 4b.

With reference to FIGURE 4, it will be seen that signal S111 is fed toboth the summing inputs 44 and 43 for circuits N1 and N2; and the outputS01 is fed forwardly by conductor 46 to summing input 43 for circuit N2.Thus summing input 44 for the rst circuit N1 has fed to it only theinput signal S1n and the bias signal SBI. On the other hand summinginput 43 for the second device N2 has fed to it not only the inputsignal S111 and its bias SB2, but also the output S01 of device N1. As aresult the output wave form, shown in FIGURE 4b, has a horizontal offsetcontrolled by bias SB2 and thereafter increases proportionately to theinput S111 at one-half the rate of slope of the output S01 up to thevalue represented by bias SB1, at which point it decreases at the samerate because of the subtraction of output S01 at the input 43.

The next wave form S03 shown in FIGURE 4c is generated by device N3, andit will be observed that the summing input 47 for device N3 is connectedto all of the preceding signals viz. Sin, S01, and S02 as Well as itsown bias signal SB3. The derivation of output signal S04 shown in FIGURE4d is obtained in a similar manner from device N4. The summing input 48for device. N4 is connected to all of the preceding signals, viz. S111,S01, S02, and S03 as Well as its own bias signal SB4.

As will be observed from the foregoing there is thus derived a series ofsymmetrical half-wave rectified triangular wave functions having theseries relationship hereinabove defined. With reference to FIGURES 2a-e,it will be noted that the base break points of the several wavefunctions progress l, 2, 4, 8, and that this indexing of the severalfunctions is controlled by the feed forward and bias method depicted inFIGURE 4. The outputs S01, S02, S03 and S04 of the several devices, N1,N2, N3 and N4 may be fed to a summing output 57 to provide a segmentalapproximation of a parabolic (square-law) function; or device N4 may beadjusted to provide a full-wave output which then may be converted intoan error curve function and then summated with outputs S01, S02 and S03as hereinabove described.

In accordance with the present invention the several devices N1, N2, N3and N4 each inclu-de an electronic network centering about anoperational amplifier (see operational amplifiers 1, 2, 3, 4, 5, 6, 7and 8 in FIGUR-E An operational amplifier is a direct coupled high gainamplifier of huge negative gain as indicated by the minus infinity signinside the standard block envelope for the amplifier. Alsocharacteristic of the use of such an amplifier because of its infinitegain, is of the large amount of feedback from output to input whichnormally is so large that the behavior of the circuit is described bythe components feeding the amplifier and placed around it to provide thefeedback. With reference to amplifier 1 in FIG- URE 5 it will be notedthat resistors R1, R2 and R3 are connected in the input to the amplifierand resistor R25 is connected in the feedback path. Also characteristicof the operational amplifier, the input is held at a virtual groundsince even a very small input voltage will drive the amplifier tosaturation. Accordingly, the gain is a function of the ratio of theinput and feedback resistors. If these are made equal the gain is minusl and if the ratio is changed to say 10, the. gain would be minus 10.While the ideal operational amplifier has a gain of minus infinity, acommercial operational amplifier may have a gain of about 100,000 whichfor practical purposes is equivalent of infinite. For example, in suchan amplifier one millivolt input will cause a full swing volt output.Since operational amplifiers are well understood in the art furthervdetails of construction are not required to be given here and the usualblock diagram as used in FIGURE 5 will sufiice.

summarizing in part the foregoing and with reference also to FIGURES 4and 5 the input signal S111 for amplifier 1, see FIGURE 5, is appiied toresistor R1 and to the input terminal S6. The output terminal 57 of theamplifier is connected to the input terminal 56 by a feedback loopincluding diode CR1 and resistor 2S, the diode CR1 being connected topass current only when the amplifier output is positive. Importantlyalso there is connected between the input and output terminals 56 and 57a second feedback loop including diode CR2 which is connected to passcurrent only when the output terminal 57 is negative and block feedbackwhen the output voltage at 57 is positive. The output terminal for thenetwork is taken at point 58 in the first mentioned feedback loopbetween diode CR1 and resistor R25. Output voltage S01 (with referencealso to FIGURE 4) appears at this point.

The operational amplifier and its network thus has (a) a conductingmode; and (b) a non-conducting mode. In the conducting mode:

In the non-conducting mode:

Accordingly, when the input voltage S1n increases negatively, the outputvoltage S01 appears as a fiat ramp as seen in FIGURES 4a and 2d; andwhen the input voltage Sm is positive, the output voltage S01 is zero,thus accomplishing the half-wave rectification hereinabove discussed.

With reference to FIGURE 5 and to operational amplifier 1, the linearramp output S01 is shifted by reference voltage Sbl. As will 4beobserved a constant positive 100 volt reference voltage is provided atthe terminal 61 of a voltage divider circuit including series connectedresistors R19, R21, R22, R23, R24 and ground 62. For purposes of presentillustration the bias voltage SBI is set at +32 volts as derived fromterminal 63 in the voltage divider circuit and resistor R2.Consequently, the diodes in the first circuit constrain the output S01to zero except when the input signal is greater than 32 volts. In otherwords so long as the input signal Sin is positive no output voltage willappear. As S111 goes negative no output voltage will appear until S111reaches and exceeds minus 32 volts. At that point the lineal ramp outputS01 of FIGURES 2d and 4a will commence. As an important feature of thepresent invention this break point or corner is very sharp-much sharperthan can be obtained with passive diode networks.

Operational amplifier 2 and its surrounding network is essentiallysimilar to that described in connection with operational amplifier 1with the following important changes:

(a) The bias voltage SBZ is reduced one-half so that the first breakpoint of the triangular wave to be formed will be located at 11:4instead of 11:8 as illustrated in FIGURES 2c and 2d; and

(b) The signal input and feedback resistors R4 and R26 and the feedforward input resistor R5 are adjusted to provide relative slopes ofS1n=+l and S01=-2.

As a result of the foregoing adjustments output signal S07I will remainat zero as the input signal goes negative to 16 volts and then willincrease as a lineal ramp until the input signal reaches minus 32 volts.At that point the output signal S01 of the first circuit appears also atthe input terminal 64 of the second circuit by reason of a forwardconnection made by conductor 66 which leads from output terminal 58 ofthe first circuit forwardly through resistor R to the input 64 of thesecond circuit. Resistor R5 adjusts the gain of output S01 to two timesthe gain of the second circuit. From this point on the voltage fedforward from the first circuit will increase at twice the rate as thevoltage S111 applied to the second circuit and at opposite polaritybecause of the reversal in the polarity produced lby operationalamplifier 1. Accordingly, as the signal reaches minus 32 volts,corresponding with 11:8 in FIGURE 2, the ramp turns down as seen at 67in FIGURE 2c. When the output S02 has been driven down to zero by theforward fed signal S01, diode CR3 holds the output at zero to the end ofthe scale, that is, between 11:12 and 11:16 as shown in FIGURES 2c.

Operational amplifier 3 and its network again is essentially similar tothat shown in circuits 1 and 2 with the following modication: the biasvoltage SB3 is again reduced one-half to eight volts so that the ramp ofthe output signal S02 of circuit number 3 will commence at 11:2 insteadof 11:4 as shown in FIGURES 2b and 2c.

The output signals S01 and S02 of lthe first two circuits are both fedforward by conductors A66 and 68 and are applied through resistors R9and R10 respectively to the input terminal 69 of operational amplifier3.

As a result of the foregoing signal S03 will be zero as input signalS111 changes from zero to minus 8 (corresponding with n=2 in FIGURE 2b)and S03 will start up` as a ramp as the input signal S111 goes fromminus 8 to minus 16 (corresponding with 11:4 in FIGURE 2b). At thispoint the output signal S02 from the second circuit is received at theinput 69 of the third circuit and, as above explained, signal S02increases at twice the rate and at reverse polarity as compared to thesignal S111 of the third circuit. Accordingly, when the input signal Sreaches minus 16 volts (the bias of the second circuit and correspondingwith 11:4) the output ramp S03 turns downwardly as seen at 71 in FIGURE2b until it is driven to zero by signal S02 and it will remain at zeroas signal S02 continues to increase to a point corresponding with 11:8.At this point, signal S01 is impressed at the input terminal 69 of thethird circuit through R9 with a relative slope of +2 and signal S02reverses direction to a relative slope of 2, so that the net slope ofthese two feedforward signals is zero. The slope of the output signalS03 at point 11:8, therefore, is determined solely by the input signalS111 with a relative slope of -1. The output S02 remains clamped at zerountil the net sum of all three input voltages reaches zero potential atn=10. At this point, the output becomes positive and begins to risepositively from zero to a second peak value at 11:12 as S111 increases.At 11:12, the feedforward voltage S02 reaches Zero so that the net inputvoltage, consisting of S111 with a relative slope of -l and thefeedforward voltage S01 with a relative slope of +2, causes the outputvoltage to reverse direction with a negative slope of 1. Voltage S03decreases until it reaches a value of zero at 11:14 and remains clampedat zero even though both S01 and S111 continue to increase to 11:16.

Operational amplifier 4 and its network, which produce the full-wavetriangular output used to drive the errorcorrection diode network, issimilar to the circuits described for amplifiers 1 through 3 with thefollowing eX- ceptions:

(a) The signal input and feedback resistors R13 and R28 and thefeedforward input resistors R14, R15, and R16 (for S01, S02, and S03,respectively( are adjusted to produce relative slopes of S111:1, 803:-2,S02=-2, 801:-2; and v (b) the rectifying circuitry, consisting of theclamping diodes in the feedback path of the amplifier, is eliminated sothat a full wave output is produced; and

(c) the bias voltage SB4 is again reduced, but this time to a smallervalue than the four volts which might be expected for the fourthcomponent waveform. Since the fourth amplifier generates the drivingwaveform for the error correction curve in order to represent, in thisexample, a six-component simulation of the parabolic curve, or 26:64segment curve, the initial bias of the fourth waveform would then be thesame as for a hypothetical sixth waveform, or one volt (%4 of the totalrange of 64 volts assumed for S111 in the present illustration).

As a result of the foregoing, the output signal S04 will be zero forvalues of S111 between zero and minus one volt (corresponding to 11:1/4That is, bias voltage SB4 is selected to adjust S04 to approach a fullwaveform shown by solid line 42 in FIGURE 3, for the generation ofparabolic error curve 41. When S111:minus 1 volt, the negative inputsignal overtakes the positive one volt bias and the output S04 begins toincrease lineally with a relative slope of +1 as S111 is furtherincreased. When S111 equals -8 volts at 11:2, the output voltage S03which commences at that point, is fed forward through conductor 73 andresistor R16 with a relative slope of 2, causing the output S04 tochange directions and decrease with a relative slope of minus 1. S04 hasa value of Zero volts at n=4. At this point, the feedforward voltage S02changes to a relative slope of +2 but its effect is cancelled out by thefeedforward voltage S02 which commences at that point `with a slope of2. As a result, output S04 is determined only by the input signal S111which causes it to Bias correction Up to this point in the discussion ofthe generation of the triangular waveforms S01, S02, S03 and S04, thespecial bias of one-half unit necessary to convert the general paraboliccurve F(n):1/2(112-n) to the square-law function F(n+1/2):1/2 (m2-1A)has been disregarded. Since it is desired to shift the overall curveone-half unit to the left to accomplish the desired transformation, thismay be done by shifting each component waveform one-half unit to theleft by reducing the positive lbias. In the 64- segment approximation,with a 64 volt range of input, as in the present example, a unit isdefined as 64 volts total range of input divided by 64 segments, or onevolt, which is equal to 11:11 with reference to FIGURES 2 and 3.Therefore, a negative bias of n:1/2 %:1/s must be added to each waveformso that SB1:77/s units (+311/2 volts), SB2:3% units (151/2 volts),SB3:17; units (+71/z volts), and SB4:1; unit (+1z volt).

The error correction curve, FIGURE 3, is generated by a conventionalfunction generator of the passive diode network type, represented byblocks +D and -D in FIGURE 5. With reference to block +D, it will beseen that the output S04 of circuit number 4 is applied to the inputterminal of the passive diode network made up of seven sections 77, 78,79, 80, 81, 82 and 83, each consisting of a resistor and diode asillustrated. These sections are connected at spaced voltage points to avoltage divider 84 connected at one end to input terminal 76 and at itsother end to a reference voltage as for example minus volts as hereshown. The several sections 77-83 are connected to a common outputterminal 86.

As an important feature of the present passive diode network, an eightsegment square-law wave form is generated by a seven-branch network foreach half cycle or full excursion of the input triangular wave formi.This is true since the first diode path does not conduct at the zeroinput level. Since the passive network provides an eight sectionsegmental approximation of a parabolic function,

11 it serves in effect to add three additional circuits of the type ofcircuits 1, 2 and 3 hereinabove described. The result is the provisionof a segmental approximation of a parabolic function having `64 sectionswhich approaches the optimum square-law function.

It is possible to make a fairly accurate sixteen section passivenetwork. With such a sixteen section network it would not be necessaryto come down to the fourth amplifier. One could use the third amplifierconverted to full wave as above discussed and drive the sixteen sectionpassive network with two peaks; and summate the result with the firsttwo amplifiers in the manner herein descri-bed, and still get a 64section segmental approximation of the desired parabolic function.

Description of overall apparatus The overall schematic diagram of thequarter-square electronic analog multiplier is illustrated in FIGURE 5.The primary input circuitry is the same as that for existingquarter-square multipliers. The two input signals X and Y are appliedsimultaneously as two-phase signals ('-j-X and -X, -i-Y and --Y) to thepositive input summing and absolute value network y(Block I-j-A) and anegative input summing and absolute value network (Block A). The outputof the network I-I-A is a positive quantity X -Y regardless of thepolarities of the individual quantities X and Y. Similarly, the outputof network -A is always a negative quantity (X-j-Y).

These inputs are applied to an active positive or negative squaringnetwork (Block `-l-B or -B). Each active network functions electricallyin accordance with the description applicable to FIGURE 4. Withreference to the positive squaring network (Block `-l-B), the junctionof resistors R1, R2, and R3 is equivalent to summing point S1 in FIGURE4. The negative signal (X-l-Y) from input lblock -A is equivalent to thecommon input signal Sin and is applied to all four stages of the activenetwork through R1, R4, R8, and R13. The bias signals corresponding toSB1, SBZ, etc. In FIGURE 4 are derived from a positive voltage divider(R19 through R24) and are applied to all four stages of the activesquaring network through R2, R6, R11, and R17. The feedforwardconnection from the output of one stage to the input of each successivestage is accomplished from stage 1 through RS, R9, and R14, from stage 2through R10 and R15, and from stage 3 through R16.

An additional input is applied to each stage through R3, R7, R12 and R18for diode compensation as discussed below.

Summary of function of the active squaring network stage With referenceto FIGURE 5, amplifier No. 1 with the immediately connected passivecomponents, functions as the generator N1 of triangular waveform S01 inthe description applicable to FIGURE 4. Amplifier 1 characteristicallyinverts the output signal polarity -with respect to the input polarity.Rectifier CR'Z holds the output signal level at zero as long as the sumof input signals has a positive polarity. Rectifier CR1 switches in theamplifier feedback component R to initiate the normal summing functionwhenever the sum of input signals has a negative polarity. The action ofthese two elements provide the rectifying and biasing requirements forthe stage. The gain of the stage, which determines the slope of theoutput signal relative to the input signal is controlled by the basicrelationship of DC amplifier operation: gain: the ratio of feedbackresistance to the individual input resistance. Two separate requirementsfor the slope of the individual waveforms were previously described: (l)to generate a series of triangular waveforms, the iirst waveforms in theseries must be fed-forward to the successive waveform-generating unitswith a specific slope relationship, and (2) the complete series ofindividual waveforms must be given a different slope relationship andadded together to produce the overall function curve.

Requirement (l) is fulfilled by the amplifier gain determined by theinput resistors of amplifiers 1 through 4 (Block .-l-B) or amplifiers 5through 8 (Block -B). Requirement (2) is met by the gain determined bythe input resistors of the output summing amplifier 9.

Diode compensation at the input One source of error in the conventionalinput network (Block A or A) is the finite impedance of the diodes inthe absolute value circuit and their non-linear operatingcharacteristics. The voltage drop across these diodes may be treated asan error signal opposite in polarity t0 the desired input signal. Tocompensate for this undesirable signal, a voltage of equal amplitude isdeveloped across the diode CR3 in the diode compensation network (BlockC or -C). This voltage is added into the summing junction of thefollowing amplifiers to cancel out the error voltage created across thediodes in Block A or A.

The output terminals 58, 89, and 86 of the first three operationalamplifier stages and the passive squaring network stage are connectedthrough resistors R31, R32, R33 and R34 to a common output terminal 92.In a like manner the outputs of operational amplifiers 5, 6 and 7 shownin Block -B and the output of passive squaring network No. II shown inBlock -D are connected to a common output terminal 93. The two outputsof the two systems are connected from terminals 92 and 93 to the inputterminal 94 of operational amplifier 9. The product KXY, K being aconstant, appears at the out-put 96 of operational amplifier 9.

I claim:

1. The method of generating from a time variable input signal an outputsignal having 2P linear segments defining a segmental approximation of aparabolic function of the input signal which comprises, forming theinput signal into a plurality (p) of symmetrical half wave rectifiedtriangular wave signals having a series relationship in which successivetriangular signals have a frequency progression of 2j where j representsthe series 0, 1, 2, 3 p-l and with the base break points of eachtriangular signal coinciding with the peaks of the next higher orderfrequency signal, and summing said triangular signals and adjusting thepeak magnitudes thereof to selected values producing said segmentaloutput signal having a constant change in slope between successivesegments.

2. The method of squaring a time variable quantity which comprises,representing the quantity to be squared as the magnitude of a timevariable electrical input signal, forming the input signal into a seriesof symmetrical half wave rectified triangular wave signals having afrequency progression of 2j where y' represents the series 0, 1, 2, 3and with the base break points coinciding with the peaks of the nexthigher order frequency signal, summating said triangular signals toproduce a segmental wave form, adjusting the successive peak magnitudesof said triangular wave signals to produce a change in slope insuccessive segments of said segmental wave form wherein sa1d change inslope is a constant approximating a parabolic wave form having thegeneral formula Ax2 plus Bx plus C, and biasing the base break points ofsaid triangular wave signals with respect to the input signal toeliminate the factor Bx.

3. The method of squaring a time variable quantity which comprises,representing the quantity to be squared as a time variable voltage,impressing said voltage as an input to an electronic network having yaplurality (p) of voltage outputs all being functions of said inputvoltage and being a series of symmetrical half wave rectified triangularsignals having a frequency progression of 2j and with the base breakpoints of each triangular signal coinciding with the peaks of the nexthigher order frequency signal, summating said outputs, and adjusting thesuccessive peak magnitudes of said triangular signals to where the firstthree terms of the series represent the amplitudes ofthe triangularsignals for j equals 0, i equals l, j equals 2 respectively and lastterm is a general formula defining the amplitude of the fm triangularsignal for values ljjmx-l.

and where:

K equals an `arbitrary constant describing the peak."

amplitude of the triangular signal for j equals 0, jmx equals the totalnumber of triangular signals used.

and with the base break points of each triangular signals coincidingwith the peaks of the next higher order frequency triangular signal andsummating said triangular signals.

5. An apparatus for squaring a time variable quantity represented 'by atime variable input voltage signal comprising, a non-linear electronicnetwork having an input adapted to receive such voltage signal andhaving a plurality (p) of outputs producing as a function of the inputvoltage signal a series of symmetrical half wave rectified triangularwave signals having a frequency progression of 25 where j represents theseries 0, l, 2, 3 and with the base breakpoints of each triangularsignal coincidingv with` the peaks of the next higher order frequencytriangular signal and means connected to said outputs summing saidtriangular signals and adjusting the successive peak magnitudes of saidtriangular signals to produce a segmental parabolic function having 2psegments having a constant change in slope between successive segments.

6. An apparatus as defined in claim 5 wherein said electronic networkincludes a plurality of operational amplifiers having feedback diodesconnected in the circuit thereof providing said non-linear electricalcharacteristics of said network.

7. An apparatus for squaring a time variable quantity represented by theamplitude of a time variable input voltage signal comprising; anelectronic network having an 'input adapted for connection to thevoltage signal, a plurality of operational amplifiers each having aresistive diode feedback circuit providing an output, and inputresistors connected to receive the input signal and being connected tothe outputs of certain other said amplifiers providing at said outputs aseries of symmetrical half wave rectified triangular wave signals havinga frequency progression of where j represents the series 0, 1, 2, 3 andwith the base break points of each triangular signal coinciding with thepeaks of the next higher frequency signal; a passive diode squaringnetwork h aving an output; and means summating the signals appearing atsaid squaring network output and said electronic network outputs exceptthe highest frequency signal output thereof and adjusting the magnitudesof the summated output signals to produce a segmental parabolic functionhaving a constant change in slope between successive segments.

;8. An apparatus for generating an Output signal as a segmentalapproximated parabolic function of a time variable input signalcomprising, an electric summing circuit each having a plurality ofinputs and an output equal to the summation of said inputs, a pluralityof non-linear function generating electric circuits each having inputand output'relationships as follows:

if input S 0, output equals -CS if input SO, output equals 0 where S isan electrical signal and C is a constant, the output of'each summatingcircuit being individually connected to the input of a separatenon-linear circuit providing a'plurality of associated circuits, saidassociated circuits being arranged to provide an inter-connected seriesthereof in which the non-linear circuit output of each associatedcircuit is connected to one of the summing circuit inputs of allsucceeding associated circuits of the series, another of the inputs ofeach summing .circuit being connected to receive the input signal,biasing means being connected one to an input of each of said summingcircuits providing biasing signals thereto progressively decreasing inmagnitude by a factor of l in each successive associated circuit of theseries, and a summing circuit connected to the outputs of all of thenon-linear circuits, whereby said circuits co-function with theaforesaid constant C providing the output signal as a plurality ofcontinuous electrical segments having a constant change in slope betweensuccessive segments.

9. An apparatus as defined in claim 8 wherein each of said associatedcircuits comprise, an operational amplifier having a feedback networkconsisting of a serially connected resistor and diode providing a firstfeedback path and a diode oppositely poled with respect to said firstnamed diode providing a second feedback path.

10. An apparatus for generating a segmental voltage signal approximationof a parabolic function in response to a time variable input voltagesignal comprising, a plurality of electric summing circuits each havinga plurality of inputs and an output equal to the summation of saidinputs, a plurality of non-linear function generating electric circuitseach having input and output relationships as follows:

if input S 0, output equals -CS if input S50, output equals 0 where S isa voltage signal and C is a constant, the output of each summing circuitbeing individually connected to the input of a separate non-linearcircuit providing a plurality of n associated circuits, said nassociated circuits being arranged to provide an interconnected seriesthereof defined by the non-linear circuit output of each associatedcircuit being connected to one of the summing circuit inputs of allsucceeding associated circuits of the series, one of the inputs of eachsumming circuit being connected to receive the input signal, a pluralityof fixed voltage biasing means being connected one to an input of eachsaid summing circuits except that of the nih associated circuit of theseries, a passive diode squaring network having an input and an outputwith the input connected to the nonlinear circuit output of the nthassociated circuit, and a summing circuit connected to the output ofsaid passive network and the non-linear circuit outputs of all of theassociated circuits except that of said nth associated circuit, wherebysaid circuits co-function with the aforesaid constant C to provide thesegmental voltage signal consisting of a plurality of continuouselectrical segments having a constant change in slope between successivesegments.

11. The method of generating an electrical output signal having aparabolic relationship with a time variable electrical input signalwhich comprises, shaping the input signal into a plurality (p) ofsymmetrical half wave rectified triangular wave signals having a seriesrelationship in which the successive signals have a frequencyprogression of 2j where j represents the series 0, l, 2, 3 p-l and withthe base break points of each triangular signal coinciding with thepeaks of the next higher frequency signal and summating the triangularsignals with successive peak magnitudes following a relationshipdefining a segmental parabolic electrical signal having a constantchange in slope between successive segments and generating a segmentalelectrical signal approximating the error deviation of said segmentalparabolic electrical signal from a parabola, and summating said errorsignal and said segmental parabolic signal to produce the output signal.

12. The method defined in claim 11 wherein a series of n triangularsignals are provided and the nth signal is shaped to provide saidsegmental error signal.

13. The method of squaring a time variable quantity which comprises,representing the quantity to Abe squared as the amplitude of a timevariable and voltage signal and impressing said voltage signal as aninput to an electronic network having a plurality (p) of voltage outputsall being functions of said input voltage and being a series ofsymmetrical half wave rectied triangular signals having a frequencyprogression of 2J and with the base break points of each triangularsignal coinciding with the peaks of the next higher frequency triangularsignal and summating said outputs except the highest order signal outputand adjusting the successive peak magnitudes of the suiimated triangularsignals to produce a segmental parabolic voltage function having 2P-1segments, applying said highest order frequency signal output to apassive diode squaring network to generate an error wave voltageapproximating the error deviation of said segmental parabolic functionfrom a parabola, and summating said error Wave voltage and saidsegmental parabolic function.

References Cited UNITED STATES PATENTS 2,900,137 8/1959 Giser 23S-1943,191,017 6/1965 Miura et al 235--194 3,253,135 5/1966 Collings et al235--194 FOREIGN PATENTS 151,873 10/ 1961 U.S.S.R.

MALCOLM A. ,MORRISON Primary Examiner R. W. WEIG, Assistant Examiner

